Creating star shapes using polar coordinates and trigonometry.
- Author
- Sukesh Ashok Kumar
This example demonstrates how to create a perfect star by combining polar coordinates, trigonometry, and symmetry. Stars are excellent for learning angle division, coordinate conversion, and line drawing algorithms.
Mathematical Concepts:
- Polar coordinates: (radius, angle) instead of (x, y)
- Angle division: 360°/n creates n evenly-spaced points
- Conversion: x = r·cos(θ), y = r·sin(θ) (polar to Cartesian)
- Radians: 2π radians = 360°, π/5 radians = 36°
- Rotational symmetry: star has 5-fold rotational symmetry
- Alternating pattern: outer points alternate with inner points
In This Example:
- 5-pointed star (classic pentagram shape)
- Outer radius: 150 pixels (tips of star)
- Inner radius: 60 pixels (indentations between tips)
- Total vertices: 10 (5 outer + 5 inner, alternating)
- Angular spacing: π/5 radians = 36° between consecutive vertices
Star Construction Algorithm:
- Place points on two concentric circles (outer and inner)
- Alternate between outer and inner radius
- Evenly distribute 10 points around the circles
- Connect consecutive points with straight lines
- This creates the characteristic star shape
Programming Concepts:
- Polar to Cartesian coordinate conversion
- Modulo operator (%) for alternating patterns
- Ternary operator: condition ? valueIfTrue : valueIfFalse
- Line drawing using linear interpolation
- abs() function from stdlib.h
- Using define for constants (PI)
What you'll learn:
- How to work with polar coordinates in graphics
- Converting between coordinate systems
- Creating symmetric shapes using trigonometry
- The relationship between angles and circle division
- Drawing lines by interpolating between endpoints
- Using modulo for alternating patterns
- Why polar coordinates simplify rotational patterns
Polar Coordinates Explained: Instead of (x,y) position, use (radius, angle):
- Radius: distance from center
- Angle: direction from center (measured from right, counterclockwise)
- To plot: x = centerX + radius·cos(angle) y = centerY + radius·sin(angle)
Line Drawing Algorithm: To draw line from (x1,y1) to (x2,y2):
- Calculate differences: dx = x2-x1, dy = y2-y1
- Determine steps: max of |dx| or |dy|
- For each step, interpolate: x = x1 + dx·step/steps
- This ensures continuous line without gaps
Experiment Ideas:
- Change points to 6: creates 6-pointed star (Star of David)
- Change points to 8: creates 8-pointed star
- Adjust innerRadius to 100: changes star sharpness
- Try innerRadius = outerRadius: creates regular polygon
Compile:
gcc m-star.c gfx/simplegfx.c -o output/m-star -lX11 -lm
Note: -lm links the math library for cos() and sin() functions
Run:
1.9.1